CSIR NET Maths previous year question papers

Single choice · The Real Line

1. A

   There exists a non-empty finite set in C

2. B

   There exists a countably infinite set in C

3. C

   There exists an uncountable set in C

4. D

   C is empty

Single choice · The Real Line

1. A

   if $liminf_{n}\to\infty a_{n} = limsup_{n}\to\infty a_{n}$, then $(a_{n})$ is convergent

2. B

   if inf{$a_{n}$ | n ≥ 1} $= limsup_{n}\to\infty a_{n}$, then $(a_{n})$ is convergent

3. C

   if sup{$a_{n}$ | n ≥ 1} $= liminf_{n}\to\infty a_{n}$, then $(a_{n})$ is constant

4. D

   if sup{$a_{n}$ | n ≥ 1} = inf{$a_{n}$ | n ≥ 1}, then $(a_{n})$ is constant

Single choice · Continuity and Differentiation

1. A

   0

2. B

   1

3. C

   Countably infinite

4. D

   Uncountable

Single choice · Continuity and Differentiation

1. A

   If f is continuous on the set S, then f is continuous on the set $\mathbb{R}$ \ S

2. B

   If g is continuous, then f is continuous on the set S

3. C

   If g is identically 0 and f is continuous on the set $\mathbb{R}$ \ S, then f is identically 0

4. D

   If g is identically 0 and f is continuous on the set S, then f is identically 0

Single choice · Sequences and Series of Functions

1. A

   There exists $x \in \mathbb{R}$ such that f(x) is not defined

2. B

   f(x) = 0 for all $x \in \mathbb{R}$

3. C

   f(x) = x for all $x \in \mathbb{R}$

4. D

   f(x) = |x| for all $x \in \mathbb{R}$

Single choice · Vector Spaces and Linear Maps

1. A

   If A is one-to-one but not onto, then m > n

2. B

   If A is onto but not one-to-one, then m < n

3. C

   If A is bijective, then m = n

4. D

   If A is one-to-one, then m = n

Single choice · Vector Spaces and Linear Maps

1. A

   There exists a vector $v \in \mathbb{R}^{10}$ such that Av ≠ 0 and $A^{2}v = 0$

2. B

   There exists a vector $v \in \mathbb{R}^{10}$ such that $A^{2}v \ne 0$

3. C

   A must have a non-zero eigenvalue

4. D

   $A^{7} = 0$

Single choice · Eigenvalues and Canonical Forms

1. A

   24 − ab = 4c

2. B

   a + b = 8

3. C

   c = 6

4. D

   ab = 0

Single choice · Vector Spaces and Linear Maps

1. A

   (x − 2)(x − 1)

2. B

   $x^{2}(x - 2)(x - 1)$

3. C

   $(x - 2)^{2}(x - 1)^{2}$

4. D

   $(x^{2} - 2)(x^{2} - 1)$

Single choice · Inner Product Spaces and Forms

1. A

   A is invertible

2. B

   There exist non-zero vectors v, w such that ⟨v, w⟩ = 0

3. C

   ⟨u, v⟩ ≠ ⟨u, w⟩ for all non-zero vectors u, v, w with v ≠ w

4. D

   Every eigenvalue of $A^{2}$ is positive

Single choice · Inner Product Spaces and Forms

1. A

   $x^{2} + 2y^{2} + 3z^{2}$

2. B

   $x^{2} + 2y^{2} + 3z^{2} - 2$xy

3. C

   $x^{2} + 2y^{2} + 3z^{2} - 2$xy − 2yz

4. D

   $x^{2} + 2y^{2} - 3z^{2}$

Single choice · The Real Line

1. A

   The series is divergent.

2. B

   The series is convergent.

3. C

   The series is conditionally convergent.

4. D

   The series is absolutely convergent.

Single choice · The Real Line

1. A

   $limsup_{n} \exp(\cos((n\pi + (-1)^{n}\cdot2e)/(2n))) > 1$.

2. B

   $lim_{n} \exp(log_{e}((n\pi^{2} + (-1)^{n}e^{2})/(7n)))$ does not exist.

3. C

   $liminf_{n} \exp(\sin((n\pi + (-1)^{n}\cdot2e)/(2n))) < \pi$.

4. D

   $lim_{n} \exp(\tan((n\pi^{2} + (-1)^{n}e^{2})/(7n)))$ does not exist.

Single choice · Continuity and Differentiation

1. A

   f(x) = sin(1/x)

2. B

   f(x) = $e^{-1/x^{2}}$

3. C

   f(x) = eˣ cos(1/x)

4. D

   $f(x) = \cos x \cdot \cos(\pi/x)$

Single choice · Eigenvalues and Canonical Forms

1. A

   The eigenspace of A for the eigenvalue 0 is two-dimensional.

2. B

   All the eigenvalues of A are real.

3. C

   $A^{3} = 0$.

4. D

   A is diagonalizable.

Single choice · Eigenvalues and Canonical Forms

1. A

   Both (a) and (b) are true.

2. B

   Both (a) and (b) are false.

3. C

   (a) is true and (b) is false.

4. D

   (a) is false and (b) is true.

Single choice · Inner Product Spaces and Forms

1. A

   ⟨x, y⟩ $= \sum_{i,j=1}^n$ $x_{i}y_{j}$

2. B

   ⟨x, y⟩ $= \sum_{i,j=1}^n$ $(x_{i}^{2} + y_{j}^{2})$

3. C

   ⟨x, y⟩ $= \sum_{j=1}^n$ $j^{3} x_{j}y_{j}$

4. D

   ⟨x, y⟩ $= \sum_{j=1}^n$ $x_{j} y_{n}$₋$_{j}$₊$_{1}$

Single choice · Lebesgue Measure and Integration

1. A

   S is in bijection with the set of rational numbers.

2. B

   S is in bijection with the set of real numbers.

3. C

   S is in bijection with S × S.

4. D

   S is in bijection with the power set of S.

Single choice · The Real Line

1. A

   inf U = 5.

2. B

   inf U = 4.

3. C

   inf U = 3.

4. D

   inf U = 2.

Single choice · The Real Line

1. A

   $\limsup a_{n} = \sqrt{3}/2$.

2. B

   $\limsup a_{2n} = 1$.

3. C

   $\limsup a_{2n} = 1/2$.

4. D

   $\limsup a_{3n} = 0$.

Single choice · The Real Line

1. A

   (a) is convergent, but (b) is not convergent.

2. B

   (a) is not convergent, but (b) is convergent.

3. C

   Both (a) and (b) are convergent.

4. D

   Neither (a) nor (b) is convergent.

Single choice · Continuity and Differentiation

1. A

   F(x) has exactly one real root.

2. B

   F(x) has exactly four real roots.

3. C

   F(x) has at most two real roots.

4. D

   F(x) has at most one real root.

Single choice · Continuity and Differentiation

1. A

   f never vanishes in [0, 1].

2. B

   f has at most finitely many zeros in [0, 1].

3. C

   f has infinitely many zeros in [0, 1].

4. D

   f(1/2) = 0.

Single choice · Continuity and Differentiation

1. A

   S = ∅

2. B

   S ≠ ∅ and S is a proper subset of $(1, \infty)$

3. C

   $(2, \infty)$ is a proper subset of S

4. D

   S ∩ (0, 1) ≠ ∅

Single choice · Lebesgue Measure and Integration

1. A

   Y is an infinite set.

2. B

   Y has $2^{|X|}$ elements.

3. C

   There is a bijective function from X to Y.

4. D

   There is a surjective function from X to Y.